# Properties

 Label 304.2.i Level $304$ Weight $2$ Character orbit 304.i Rep. character $\chi_{304}(49,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $18$ Newform subspaces $6$ Sturm bound $80$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 304.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$19$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$6$$ Sturm bound: $$80$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(304, [\chi])$$.

Total New Old
Modular forms 92 22 70
Cusp forms 68 18 50
Eisenstein series 24 4 20

## Trace form

 $$18 q + 3 q^{3} - q^{5} + 8 q^{7} - 6 q^{9} + O(q^{10})$$ $$18 q + 3 q^{3} - q^{5} + 8 q^{7} - 6 q^{9} + 4 q^{11} - q^{13} - 9 q^{15} - 5 q^{17} - 6 q^{19} - 4 q^{21} + 7 q^{23} - 6 q^{25} - 18 q^{27} - q^{29} + 24 q^{31} + 2 q^{33} - 4 q^{37} + 2 q^{39} + q^{41} - 7 q^{43} + 12 q^{45} + 7 q^{47} + 2 q^{49} + 11 q^{51} + 3 q^{53} - 3 q^{57} + 19 q^{59} + 19 q^{61} - 24 q^{63} + 6 q^{65} - 19 q^{67} - 30 q^{69} - 19 q^{71} + q^{73} + 52 q^{75} - 40 q^{77} - 31 q^{79} - 17 q^{81} - 52 q^{83} + 9 q^{85} - 78 q^{87} + 3 q^{89} + 40 q^{91} - 20 q^{93} - 13 q^{95} + 17 q^{97} - 12 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(304, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
304.2.i.a $2$ $2.427$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$1$$ $$0$$ $$q+(-1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots$$
304.2.i.b $2$ $2.427$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$-3$$ $$0$$ $$q+(1-\zeta_{6})q^{3}+(-3+3\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots$$
304.2.i.c $2$ $2.427$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$0$$ $$8$$ $$q+(1-\zeta_{6})q^{3}+4q^{7}+2\zeta_{6}q^{9}-3q^{11}+\cdots$$
304.2.i.d $2$ $2.427$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$4$$ $$0$$ $$q+(1-\zeta_{6})q^{3}+(4-4\zeta_{6})q^{5}+2\zeta_{6}q^{9}+\cdots$$
304.2.i.e $4$ $2.427$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$0$$ $$0$$ $$-2$$ $$-4$$ $$q+\beta _{1}q^{3}+(-1+\beta _{1}-\beta _{2})q^{5}+(-1+\cdots)q^{7}+\cdots$$
304.2.i.f $6$ $2.427$ 6.0.2696112.1 None $$0$$ $$1$$ $$-1$$ $$4$$ $$q+(-\beta _{1}-\beta _{2}-\beta _{3}+\beta _{5})q^{3}+(\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(304, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(304, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(38, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(76, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(152, [\chi])$$$$^{\oplus 2}$$